Evaluate lim_{x→∞} (1 + 1/x)^x.

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Study for the AP Calculus BC Test. Discover flashcards and multiple choice questions with hints and explanations to prepare effectively. Ace your exam!

Multiple Choice

Evaluate lim_{x→∞} (1 + 1/x)^x.

Explanation:
This limit is a classic way to define e. To see why it tends to e, take natural logs: let L be the limit of (1 + 1/x)^x as x → ∞, and consider ln L = lim x→∞ x ln(1 + 1/x). For small u, ln(1 + u) = u − u^2/2 + O(u^3). With u = 1/x, ln(1 + 1/x) = 1/x − 1/(2x^2) + O(1/x^3). Multiply by x: x ln(1 + 1/x) = 1 − 1/(2x) + O(1/x^2) → 1. Therefore ln L = 1, so L = e.

This limit is a classic way to define e. To see why it tends to e, take natural logs: let L be the limit of (1 + 1/x)^x as x → ∞, and consider ln L = lim x→∞ x ln(1 + 1/x). For small u, ln(1 + u) = u − u^2/2 + O(u^3). With u = 1/x, ln(1 + 1/x) = 1/x − 1/(2x^2) + O(1/x^3). Multiply by x: x ln(1 + 1/x) = 1 − 1/(2x) + O(1/x^2) → 1. Therefore ln L = 1, so L = e.

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